Question

How do you solve the equation x² - 3x - 18 = 0 by completing the square?

a) x = (3 ± √21) / 2
b) x = (-3 ± √21) / 2
c) x = (3 ± 21) / 2
d) x = (-3 ± 21) / 2

Answer 1

To solve x² - 3x - 18 = 0 by completing the square, move the constant term to the other side, complete the square on the left side, then take the square root of both sides and solve for x. The correct solutions are x = 6 and x = -3, which do not match any of the given choices.

Step-by-step explanation:

To solve the equation x² - 3x - 18 = 0 by completing the square, we follow a series of steps:

1. Move the constant term to the other side of the equation: x² - 3x = 18.
2. Divide the coefficient of the x term by 2 and square it, then add that square to both sides: (-3/2)² = 9/4. Add 9/4 to both sides to get x² - 3x + 9/4 = 18 + 9/4.
3. Simplify the right side of the equation: 18 + 9/4 = 72/4 + 9/4 = 81/4.
4. The equation now reads x² - 3x + 9/4 = 81/4. The left side is a perfect square trinomial: (x - 3/2)² = 81/4.
5. Take the square root of both sides: x - 3/2 = ±√(81/4).
6. Solve for x: x = 3/2 ± √(81/4) = 3/2 ± 9/2.
7. The solutions are x = (3/2 + 9/2) and x = (3/2 - 9/2), which simplify to x = 6 and x = -3, respectively.

The answer is x = 6 and x = -3, which means that none of the options (a), (b), (c), or (d) provided is correct.